Method of fabricating a reflecting mirror

ABSTRACT

A method of fabricating a reflecting mirror, wherein mirror segments are fused together thereby composing the reflecting mirror comprising, formulating square sums of displacements at a plurality of sampling points on a mirror surface of the reflecting mirror as a function of a thermal expansion coefficient-vector having components of deviations of thermal expansion coefficients of the respective mirror segments from the average thermal expansion coefficients of all the mirror segments, positions of the components corresponding to arranging positions of the respective mirror segments; generating a stochastic process wherein the smaller the square sum of the displacement of the thermal expansion coefficient vector, the larger the probability whereby the thermal expansion coefficient vector appears, by a computer using random numbers; selecting the thermal expansion coefficient vector minimizing the square sum of the displacement from the appeared thermal expansion coefficient vectors; and arranging and fusing together the mirror segments in accordance with the components thereof.

This application is a continuation, of application Ser. No. 07/914,436,filed Jul. 15, 1992 now abandoned.

BACKGROUND OF THE INVENTION

1. Field of The Invention

The present invention relates to a method of fabricating a reflectingmirror wherein, when a reflecting mirror utilized in, for instance, areflecting telescope, is composed by fusing together mirror segments,the mirror segments are arranged and fused together so that the thermaldeformation caused by the difference in the thermal expansioncoefficient among the component mirror segments, is minimized.

2. Discussion of Background

FIG. 6 is a perspective view showing a conventional reflecting mirror.In FIG. 6, a reference numeral 1 designates a reflecting mirror, whichis composed by fusing together a plurality of hexagonal mirror segments(hereinafter, stacks) 2. The reflecting mirror is formed by determiningthe arrangement of the stacks 2 by intuition in accordance withindividual cases, and by fusing them together. Furthermore, a surface ofthe reflecting mirror 1 is polished to form, for instance, a paraboloidor a hyperboloid with an accuracy of about 1/100 of the observedwavelength, so that visible light, or an electromagnetic wave such asinfrared rays emitted by a celestial body, is reflected and focused.

When the surface of the reflecting mirror 1 is provided with a completeparaboloid or the like, the incident electromagnetic wave from thecelestial body geometrically converges into a point (focus). Actually,the diameter of the image of the celestial body is not nullified due tothe diffraction phenomena of light. There is a theoretical limitdetermined by the aperture D of the reflecting mirror 1 and thewavelength λ of the incident electromagnetic wave.

This theoretical limit FWHM (Full Width at Half Maximum), is generallyexpressed as follows. ##EQU1##

This theoretical limit is a width of an intensity distribution of lightwherein the intensity becomes a half of the maximum intensity as shownin FIG. 7. Accordingly, the theoretical limit in the size of the imageof a star is determined by the aperture D of the reflecting mirror 1 andthe wavelength λ of the incident electromagnetic wave. The larger theaperture D, the smaller the theoretical limit. Accordingly, increase inthe aperture of the reflecting mirror 1 enables reduction in size of theimage and hence is a significant contribution to the improvement ofresolution, the improvement of limit of detection and reduction inexposure time.

However, since thermal expansion coefficients of the stacks 2 areactually not zero, the reflecting mirror 1 suffers thermal deformationwhen the temperature thereof changes. When the thermal expansioncoefficients of the respective stacks 2 are equal, the respective stacks2 deform in similar figures. Accordingly, only the focus position of thereflecting mirror 1 moves, and an image formation accuracy thereof isnot deteriorated. However, in practice, the stacks 2 differ from eachother in thermal expansion coefficient, so that the reflecting mirror 1is subject to irregular thermal deformation. When the aperture of thereflecting mirror 1 is large, since the number of the stacks 2increases, the deformation becomes more complicated and the deformationquantity is enlarged even by a little inclination.

Accordingly, when such thermal deformation is caused, the light incidentfrom the celestial body scatter as shown in FIG. 8. The image of thecelestial body is provided with an intensity distribution as shown inFIG. 9, and becomes a dim image. Therefore, even when the aperture ofthe reflecting mirror 1 is enlarged, the advantage of reducing thetheoretical limit, can not be realized.

As major causes of the nonuniformity of the thermal expansioncoefficients of the stacks 2 which causes the nonuniform thermaldeformation, a difference in gradients of the thermal expansioncoefficients of the respective stacks 2 in the thickness directionsthereof (which causes a bimetallic deformation) and dispersing of meanthermal expansion coefficients of the respective stacks 2, are pointedout. As a method of suppressing the thermal deformation as much aspossible, a stack arrangement as shown in FIG. 10, is proposed (a firstconventional example).

In FIG. 10, variables Δα₁, . . . , Δα₃₇ attached to the respectivestacks 2, respectively designate deviations of the mean thermalexpansion coefficients of the respective stacks 2 from a mean value ofthe thermal expansion coefficients of all the stacks 2 (hereinafter,thermal expansion coefficient), which are classified into three groups(netting of crossing oblique lines, netting of dots, and withoutnetting) in an order of size of the thermal expansion coefficients (Δα₁≧Δα₂ ≧. . .≧Δα₃₇).

In this method, around the stacks 2 belonging to a group of largethermal expansion coefficients, the stacks 2 belonging to a group ofmedium thermal expansion coefficients, or a group of small thermalexpansion coefficients, are arranged. In this way, a large thermalexpansion of the stacks 2 belonging to the group of large thermalexpansion coefficients, is alleviated by a small thermal expansion ofthe surrounding stacks 2, by which the deformation becomes local, and itis expected by intuition that the deformation quantity becomes far moresmaller than in the case wherein the distribution is deviated.

FIG. 11 is a sectional diagram of a reflecting mirror provided withactuators for correcting the thermal deformation (a second conventionalexample), wherein a reference numeral 1 designates the reflectingmirror, 3, a temperature sensor attached to the backface of thereflecting mirror 1, 4, a processing unit for calculating a correctiveforce based on a measured value of a temperature of the reflectingmirror 1 obtained by the temperature sensor 3, 5, an actuatorcontroller, and 6, the actuators for correcting the thermal deformationby applying the corrective force on the reflecting mirror 1.

In this example, when the thermal deformation is to be corrected, if oneintends to totally correct it, it becomes necessary to correct evenirregularities having small pitches, which requires a large correctingforce and is not practical. Therefore, the deformation is expanded intoa series of finite terms or infinite terms which is a function ofspatial frequencies. A correction is performed by choosing terms thereofhaving large pitches of irregularities. At this moment, irregularitieshaving small pitches which remain uncorrected, become a residualdeformation of a mirror surface thereof which deteriorates the qualityof the image.

FIG. 12 shows an arrangement of the stacks 2 wherein the thermaldeformation is predicted by intuition to concentrate on the terms havinglarge pitches of irregularities, when only the terms having largepitches of irregularities, are to be corrected. In FIG. 12, thedefinition of variables Δα₁, . . . , Δα₃₇ attached to the respectivestacks 2, is the same as in the case of FIG. 10.

Since the conventional reflecting mirror is constructed as above, thearrangement of the respective stacks 2 is performed by intuition.Therefore, the arrangement is not necessarily the one for minimizingthermal deformation.

SUMMARY OF THE INVENTION

It is an object of the present invention to provide a method offabricating a reflecting mirror which minimizes thermal deformation ofthe surface of the reflecting mirror.

It is an object of the present invention to provide a method offabricating a reflecting mirror which minimizes the residual thermaldeformation remaining in the reflecting mirror after correcting thethermal deformation by expanding the thermal deformation in a series asa function of spatial frequencies.

According to a first aspect of the present invention, there is provideda method of fabricating a reflecting mirror, wherein mirror segments arefused together thereby composing the reflecting mirror comprising:

formulating square sums of displacements at a plurality of samplingpoints on a mirror surface of the reflecting mirror as a function of athermal expansion coefficient vector having components of deviations ofthermal expansion coefficients of the respective mirror segments fromthe average thermal expansion coefficients of all the mirror segments,positions of the components corresponding to arranging positions of therespective mirror segments;

generating a stochastic process wherein the smaller the square sum ofthe displacement of a thermal expansion coefficient vector, the largerthe probability whereby the thermal expansion coefficient vectorappears, by a computer using random numbers;

selecting the thermal expansion coefficient vector minimizing the squaresum of the displacement from the appeared thermal expansion coefficientvectors; and

arranging and fusing together the mirror segments in accordance with thecomponents thereof.

According to a second aspect of the present invention, there is provideda method of fabricating a reflecting mirror, wherein mirror segments arefused together thereby composing the reflecting mirror composing:

expanding a thermal deformation of the reflecting mirror in a serieshaving finite terms which is a function of spatial frequency;

formulating a square sum of components of a residual deformation vectorsignifying a residual deformation quantity after correctingpredetermined terms of the finite terms as a function of a thermalexpansion coefficient vector having components of deviations of thermalexpansion coefficients of the respective mirror segments from theaverage thermal expansion coefficients of all the mirror segments,positions of said components corresponding to arranging positions of therespective mirror segments;

generating a stochastic process wherein the smaller the square sum ofthe components of the residual deformation vector, the larger theprobability whereby the thermal expansion vector appears, by a computerusing random numbers;

selecting the thermal expansion coefficient vector minimizing the squaresum of the components of the residual deformation vector from theappeared thermal expansion coefficient vectors; and

arranging and fusing together the mirror segments in accordance with thecomponents thereof.

The method of fabricating a reflecting mirror according to the firstaspect of the present invention, considerably reduces the thermalexpansion quantity of the mirror surface of the reflecting mirror, bygenerating the stochastic process wherein the smaller the square sum ofthe deformations of the mirror face of the reflecting mirror, the largerthe probability whereby the thermal expansion coefficient vectorappears, by a computer using random numbers, by selecting the thermalexpansion coefficient vector minimizing the square sum of thedisplacement among these, and by arranging and fusing together thestacks in accordance with the components.

Furthermore, the method of fabricating a reflecting mirror according tothe second aspect of the present invention, considerably reduces theresidual deformation quantity remained after the correction of themirror surface of the reflecting mirror, by generating the stochasticprocess wherein the smaller the square sum of the components of theresidual deformation vectors of the surface of the reflecting mirror,the larger the probability whereby the thermal expansion coefficientvector appears by a computer utilizing random numbers, by selecting thethermal expansion coefficient vector minimizing the square sum of thedisplacement among these, and by arranging and fusing together thestacks in accordance with the components.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a plan view showing arranged positions of stacks of areflecting mirror according to the first and the second aspects of thepresent invention;

FIG. 2 is a flowchart showing an embodiment of a method of fabricating areflecting mirror according to the first aspect of the presentinvention;

FIG. 3 is a plan view showing contour maps in respective modes of thenatural vibration mode of the reflecting mirror in FIG. 1;

FIG. 4 is a flowchart showing an embodiment of a method of fabricating areflecting mirror according to the second aspect of the presentinvention;

FIG. 5 is a plan view showing an example of a stack arrangementdetermined by using gradients of thermal expansion coefficients ofstacks in the thickness directions according to the first and the secondaspects of the present invention;

FIG. 6 is a perspective view showing construction of the conventionaland the invented reflecting mirror;

FIG. 7 is a characteristic diagram showing an intensity distribution ofan image of a celestial body when there is no thermal deformation in areflecting mirror;

FIG. 8 is a sectional view of the reflecting mirror when it is thermallydeformed;

FIG. 9 is a characteristic diagram showing the intensity distribution ofthe image of the celestial body when there is a thermal deformation inthe reflecting mirror;

FIG. 10 is a perspective view showing a stack arrangement in theconventional reflecting mirror;

FIG. 11 is a sectional view showing a conventional reflecting mirrorprovided with actuators for correcting the thermal deformation; and

FIG. 12 is a plan view showing another stack arrangement in anotherconventional example.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS EXAMPLE 1

An embodiment of the first aspect of the present invention will beexplained referring to the drawings. In FIG. 1, a reference numeral 1designates a plan view showing a reflecting mirror 1 composed of 37 ofstacks 2, wherein the numbers 1, . . . , 37 attached to the respectivestacks 2 designate the positions thereof in the reflecting mirror 1.

Next, explanation will be given to the operation utilizing a flowchartof FIG. 2. First, deviations of thermal expansion coefficients of therespective stacks 2 from a mean value of the thermal expansioncoefficients of 37 of the stacks 2 to be arranged to 37 of stackpositions (hereinafter, thermal expansion coefficient) are determined asΔα₁ ≧Δα₂ ≧. . .≧Δα₃₇ in an order of size. Accordingly, determining thestack positions, signifies corresponding the deviations Δαj (j=1, . . ., 37) to the stack positions of FIG. 1. Furthermore, displacements atrespective thermal displacement measuring points (sampling points) setas they are disposed at equal intervals on the mirror surface of thereflecting mirror 1, by, for instance, 1,000 points, are defined asΔZ_(k) (k =1, . . . , 1,000).

When the thermal expansion coefficients Δαj (j=1, . . . , 37) arearbitrarily given to the respective stacks 2, the displacements ΔZ_(k)(k=1, . . . , 1,000) of the respective thermal displacement measuringpoints when temperature is changed by ΔT, can be calculated by FiniteElement Method. Accordingly, Δαj (j=1, . . . , 37) and ΔZ_(k) (k=1, . ., 1,000) have a relation as follows, utilizing a 1,000×37 matrix S whichdoes not depend on the thermal expansion coefficients Δαj (j=1, . . . ,37) of the stack 2. ##EQU2##

As shown in this equation, the fist column of the matrix S can becalculated as the displacement vector ΔZ_(k) (k=1, . . . , 1,000), when(Δα₁, Δα₂, . . . , Δα₃₇)=(0, 1, 0, . . . , 0) and the temperature changeis determined as ΔT =1° C. Similarly, the second column or the columnstherebelow can be calculated as the ΔZ_(k) (k=1, . . . , 1,000), when(Δα₁, Δα₂, . . . , Δα₃₇)=(0, 1, 0, . . . , 0), (Δα₁, Δα₂, . . . ,Δα₃₇)=(0, 0, 1, 0, . . . , 0), . . . (step ST1).

At this point, a displacement vector U and a thermal expansioncoefficient vector α are defined as follows. ##EQU3## Then, the Equation(2) can be rewritten as follows.

    U=SαΔT                                         (4)

Furthermore, the size of the deformation can be evaluated by RMS (RootMean Square) of a normal displacement, and in this examples as follows.##EQU4##

By this equation, to minimize the RMS, is to minimize the square sumΣ_(K=1), . . . , 1000 (ΔZ_(k))² of the respective displacements. Thissquare sum can be shown as follows from equations (3) and (4). ##EQU5##where the shoulder suffix (^(T)) designates a transposition of a matrixor a vector.

From Equation (6), it is found that to minimize the thermaldisplacement, is to minimize the value α^(T) S^(T) Sα. Assuming R=S^(T)S, the matrix R becomes a symmetrical matrix of 37×37 without dependingon the thermal expansion coefficient vector α (step ST2). Accordingly,the problem of the optimum arrangement becomes a problem of arrangingthe deviations Δα₁, Δα₂, . . . , Δα₃₇ in elements of the thermalexpansion coefficient vectors α so that α^(T) Rα is minimized.

Next, explanation will be given to how to arrange the thermal expansioncoefficients α so that α^(T) Rα is minimized. For this purpose, assuminga set composing a total of permutations of integer number series {1, . .. , 37}, and the set of the permutations is defined as Ω. Elements ω ofΩ designates the respective permutations. α(ω) is assumed to be athermal expansion coefficient vector α wherein components thereof arechangeably arranged by the elements ω. That is, in case of a permutationof ω as {i₁, i₂, . . . , i₃₇ }, α(ω)=(Δαi₁, Δαi₂, . . . , Δαi₃₇). Bythis definition, the problem results in finding ω which minimizes anevaluation function defined as follows.

    E(ω)=α.sup.T (ω)Rα(ω)        (7)

To find out ω minimizing the E(ω), a probability distribution isconsidered as follows.

    π(ω)=exp{-βE(ω)}/Z                     (8)

where β is a pertinent parameter having a positive value, and Z, anormalizing constant defined as follows. ##EQU6##

As is apparent in Equation (8), the smaller the value of E(ω), thelarger the probability, when ω is selected as such. The larger the valueof the parameter β , the more significant the tendency. Accordingly, ifit becomes possible to artificially generate a stochastic process (timeseries of ω) having the probability distribution as in Equation (8), itcan be expected with high probability that the value of E(ω) withrespect to the obtained sample is extremely adjacent to the minimumvalue.

The stochastic process having the probability distribution as inEquation (8), can be generated by a computer utilizing random numbers asfollows.

(1) An initial value of ω is selected at random from the set Ω ofpermutations by utilizing random numbers (step ST3).

(2) Two integers k and j are selected at random from the integer series{1, . . . , 37} by using random numbers (step ST4).

(3) An increment ΔE (ω) of E (ω) is obtained as follows when a componentof α_(k) (ω) of α (ω) is substituted by α_(j) (ω) as follows (step ST5).##EQU7##

(4) Next, a determination is performed whether α_(k) (ω) is to besubstituted by α_(j) (ω), in accordance with a probability shown asfollows.

    P (substitution is performed for α.sub.k (ω) and α.sub.j (ω))=1/{1+exp[βΔE (ω))[}           (11)

Specifically, a uniform random number r wherein 9≦r≦1, is generated(step ST6). Comparison is made between sizes of r and 1/[1+exp{βΔE (ω)}](step ST7). When r is smaller than the other, substitution of α_(k) (ω)and α_(j) (ω) is performed (step ST8). When r is larger than the other,the substitution is not performed.

(5) The operation returns to (2) and repeats the same steps.

The above steps are specified by transpositions of arbitrary twoelements, that is, a probability law of interchange. However, thetransition from an arbitrary permutation to another permutation can berepresented by a product of the interchanges. Accordingly, the generatedstochastic process goes around a total of the set Ω of permutations.After a sufficient time, the distribution of the stochastic processapproaches a stationary distribution in Ω. This stationary distributionis equal to the probability distribution of Equation (8) as shown in N.Metropolis et al, "Equation of State Calculations by Fast ComputingMachines, "J. Chem. Phys., Vol. 21, 1953, pp. 1087-1091.

In this way, a series of permutation ω whereby the probability ofminimizing the value of the evaluation function E(ω), is large, isobtained. Accordingly, among these, a permutation ω minimizing the valueof E(ω), is selected (step ST9). Finally, the respective stacks 2 arearranged and fused together in accordance with the permutation ωobtained by the above means (step ST10).

EXAMPLE 2

Next, explanation will be given to another embodiment of this inventionusing a flow chart of FIG. 4. In this example, the thermal deformationis expanded in modes. Predetermined terms are corrected. At thisoccasion, the respective stacks 2 are arranged so that the residualdeformation after the correction is minimized.

First, as in Example 1, the matrix S is calculated by Finite ElementMethod (step ST1). The relationship between the displacement vector Uand the thermal expansion coefficient vector α becomes as Equation (4).The residual deformation after the correction is the displacement vectorU subtracted by the correction quantity. Therefore, to obtain therelationship between the residual deformation and the thermal expansioncoefficient vector α, the relationship between the correction quantityand the thermal expansion coefficient vector α should be obtained. Aspecific explanation will be given to a case wherein 1st to 32nd modesof the natural vibration modes, are corrected. First, the displacementvector U can be expressed as a superposition of the natural vibrationmodes having infinite terms. The natural vibration modes can becalculated by using Finite Element Method. Examples of the deformationpattern of the natural mode are shown in FIG. 3. A deformation patternq_(m) of the mth natural vibration mode is expressed by thedisplacements q_(mi) at the same coordinate point with that of thedisplacement vector U as follows. ##EQU8##

Assuming expansion coefficients (which correspond to amplitudes of thevibration mode) as A_(m), the displacement vector U can be expressed asfollows as a superposition of the natural vibration modes. ##EQU9##

Among the series, the components up to 32nd term become the correctionquantity as follows. ##EQU10##

The expansion coefficient vector A and the matrix Q are defined asfollows (step ST21). ##EQU11##

The thermal expansion coefficient vector α and the displacement vector Uare in the linear relationship as shown in Equation (4). Also, thedisplacement vector U and the expansion coefficient A are in a linearrelationship. Therefore, the thermal expansion coefficient vector α andthe expansion coefficient A are also in a linear relationship.Accordingly, A can be expressed as follows by using a matrix P.

    A=PαΔT

or, ##EQU12##

P can be calculated as follows. The first column of P are expansioncoefficients wherein the thermal deformation (which is the same with thefirst row of S) are expanded by the natural vibration modes, when thethermal expansion coefficient vector α is assumed as (1, 0, . . . , 0).The thermal deformation is calculated by Finite Element Method. The modeexpansion is calculated by methods such as fitting by least squaremethod. The second column thereof can similarly be calculated byassuming α=(0, 1, 0, . . . , 0) (step ST22).

In this way, the correction quantity QA is related to the thermalexpansion coefficient vector α as QA=QPαΔT.

Accordingly, the residual deformation vector U_(z) can be expressed asfollows by subtracting the correction quantity from the displacementvector U. ##EQU13## where

    S.sub.z =S-QP                                              (17)

Accordingly, by using the residual deformation vector U_(z) instead ofthe displacement vector U in Example 1 and S_(z) instead of S, theoptimum arrangement can be obtained as in Example 1 (step ST23). Thetreatment of step ST3 or steps therebelow is the same as in Example 1.Therefore, the explanation will be omitted.

Furthermore, in the above Examples 1 and 2, the methods of fabricating areflecting mirror are explained wherein the thermal deformation quantityof the reflecting mirror 1 or the residual deformation after thecorrection, is minimized by using the deviations of the thermalexpansion coefficients of the respective stacks 2 as the thermalexpansion coefficient vector α. However, the gradients of the thermalexpansion coefficients of the respective stacks 2 in the thicknessdirections can be utilized as the thermal expansion coefficient vectorα, with the same effect as in the above Examples.

When the gradients of the thermal expansion coefficients of therespective stacks 2 in the thickness directions are utilized as thethermal expansion coefficient vector α, an example wherein the residualdeformation quantity after the 1st to the 32nd natural vibration modesof the thermal deformation are corrected, is minimized, by which thestacks are arranged and fused together, is as shown in FIG. 5.

In FIG. 5, the variables Δα₁, . . . , Δα₃₂, (Δα₁ ≧Δα₂ ≧. . .≧Δα₃₇)attached to the respective stacks 2, show the sizes of the gradients ofthe thermal expansion coefficients of the respective stacks in thethickness directions.

As stated above, according to the first aspect of this invention, thestochastic process wherein the smaller the square sum of thedisplacement of the surface of the reflecting mirror, the larger theprobability whereby the thermal expansion coefficient vector appears, isgenerated by a computer using random numbers, the thermalexpansion-coefficient vector which minimizes the square sum of thedisplacement among these, is selected and the stacks are arranged andfused together in accordance with the components. Therefore, thisinvention has an effect wherein the thermal deformation of thereflecting mirror can be made extremely small.

According to the second aspect of the present invention, the stochasticprocess wherein the smaller the square sum of the components of theresidual deformation vector, the larger the probability whereby thethermal expansion coefficient vector appears, by a computer using randomnumbers, the thermal expansion coefficient vector which minimizes thesquare sum of the displacement among these, is selected and the stacksare arranged and fused together in accordance with the components.Therefore, as in the first aspect of this invention, the residualdeformation quantity of the reflecting mirror after the correction canextremely be made small.

What is claimed is:
 1. A method of fabricating a reflecting mirror,wherein mirror segments are fused together thereby composing thereflecting mirror, comprising:determining an average thermal expansioncoefficient for the mirror segments; determining the deviation of eachmirror segment's thermal expansion coefficient from the average thermalexpansion coefficient; generating a plurality of different vectors byrepeatedly utilizing a stochastic process wherein the components of eachvector are the deviations such that each component corresponds with oneof said segments; creating a provisional model arrangement of thesegments for each vector, such that the position of each component ofeach vector defines the position of each corresponding segment;determining sets of sampling points on each of the model arrangements;for each of the model arrangements, evaluating the square sums ofdisplacements of the sampling points, which displacements result fromthermal expansion; selecting the model arrangement which yields theminimum square sum; and arranging and fusing together the mirrorsegments in the same arrangement as the selected model arrangement; andwherein the stochastic process is such that there is an inverserelationship between the probability that the stochastic process wouldgenerate a particular vector and the square sum of the displacementswhich would correspond to the particular vector.
 2. A method accordingto claim 11 wherein the step of for each of the model arrangements,evaluating the square sums of displacements includes finding for eachpermutation of the components ω of the coefficient vector α, theevaluation function E(ω)=α^(T) (ω)Rα(ω), in which R is a square matrixindependent of the coefficient vector, and wherein the step of selectingincludes selecting the permutation of the components ω of thecoefficient vector α yielding the minimum value for the evaluationfunction E(ω).
 3. A method according to claim 2 wherein the stochasticprocess has a probability distribution π(ω)=exp{-βE(ω)}/Z, which is usedto determine the permutation of elements, wherein β is a positiveparameter and Z is a normalizing constant defined as ##EQU14##
 4. Amethod of fabricating a reflecting mirror, wherein mirror segments arefused together thereby composing the reflecting mirror,comprising:determining an average thermal expansion coefficient for themirror segments; determining the deviation of each mirror segment'sthermal expansion coefficient from the average thermal expansioncoefficient; performing a series expansion of a thermal deformationfunction of a computer model of the mirror segments to form an expansionseries having finite terms, said terms each being a function of aspatial frequency mode of displacement of the mirror segments;generating a plurality of different vectors by repeatedly utilizing astochastic process wherein the components of each vector are thedeviations such that each component corresponds with one of saidsegments; creating a provisional model arrangement of the segments foreach vector, such that the position of each component of each vectordefines the position of each corresponding segment; determining sets ofsampling points on each of the model arrangements determining a residualdeformation vector from displacements of the sampling points resultingfrom thermal expansion of the mirror segments, the residual deformationvector including a residual deformation quantity corrected usingpredetermined terms of the expansion series; for each of the modelarrangements, evaluating the square sums of components of the residualdeformation vector; selecting the model arrangement which yields theminimum square sum; and arranging and fusing together with mirrorsegments in the same arrangement as the selected model arrangement; andwherein the stochastic process is such that there is an inverserelationship between the probability that the stochastic process wouldgenerate a particular vector and the square sum of the components of theresidual displacement vector which would correspond to the particularvector.
 5. A method according to claim 4 wherein the step of for each ofthe model arrangements, evaluating the square sums of components of theresidual displacement vector includes finding for each permutation ofthe components ω of the coefficient vector α, the evaluation functionE(ω)=α^(T) (ω)Rα(ω), in which R is a square matrix independent of thecoefficient vector, and wherein the step of selecting includes selectingthe permutation of the components ω of the coefficient vector α yieldingthe minimum value for the evaluation function E(ω).
 6. A methodaccording to claim 5 wherein the stochastic process has a probabilitydistribution π(ω)=exp{-βE(ω)}/Z, which is used to determine thepermutation of elements, wherein β is a positive parameter and Z is anormalizing constant defined as ##EQU15##